CUET PG 2026 Physics Question Paper with Solutions - Available Here

Concept:

Ampère’s Circuital Law states that the line integral of the magnetic field \(\vec{B}\) around any closed path is equal to the permeability of free space multiplied by the total current enclosed by that path.

The mathematical form of Ampère’s Law is:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]

where


\(\vec{B}\) = Magnetic field
\(d\vec{l}\) = Infinitesimal length element of the closed loop
\(\mu_0\) = Permeability of free space
\(I_{enc}\) = Current enclosed by the loop




Step 1: Identify the result of the line integral.

The line integral of the magnetic field around a closed path equals \(\mu_0\) multiplied by the enclosed current.



Step 2: Write the final value.
\[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I \]

Thus, the value of the line integral is \(\mu_0 I\). Quick Tip: \textbf{Ampère’s Law:} \(\displaystyle \oint \vec{B}\cdot d\vec{l} = \mu_0 I\). The circulation of magnetic field around a closed loop depends on the enclosed current.

Concept:

A central force is a force that is always directed along the line joining the particle and a fixed point (the center) and whose magnitude depends only on the distance \(r\).

Examples include gravitational force and electrostatic force.



Step 1: Understand torque in a central force.

Torque is given by
\[ \vec{\tau} = \vec{r} \times \vec{F} \]

In a central force field, the force \(\vec{F}\) acts along the direction of \(\vec{r}\).

Thus,
\[ \vec{\tau} = \vec{r} \times \vec{F} = 0 \]



Step 2: Apply angular momentum relation.

Since
\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]

and \(\vec{\tau} = 0\),
\[ \frac{d\vec{L}}{dt} = 0 \]

Therefore,
\[ \vec{L} = constant \]



Thus, angular momentum remains conserved. Quick Tip: For any \textbf{central force}, torque about the center is zero, so \textbf{angular momentum is conserved}.

Concept:

According to the de Broglie hypothesis, particles such as electrons exhibit wave-like behavior. The wavelength associated with a moving particle is given by
\[ \lambda = \frac{h}{p} \]

where


\(h\) = Planck’s constant
\(p\) = Momentum of the particle




Step 1: Determine the kinetic energy of the electron.

When an electron is accelerated through a potential difference \(V\),
\[ K.E. = eV \]

where \(e\) is the charge of the electron.



Step 2: Relate kinetic energy and momentum.
\[ K.E. = \frac{p^2}{2m} \]

Thus,
\[ eV = \frac{p^2}{2m} \]



Step 3: Solve for momentum.
\[ p = \sqrt{2meV} \]



Step 4: Substitute into de Broglie equation.
\[ \lambda = \frac{h}{p} \]
\[ \lambda = \frac{h}{\sqrt{2meV}} \]

Thus, the de Broglie wavelength is
\[ \lambda = \frac{h}{\sqrt{2meV}} \] Quick Tip: For an electron accelerated through potential \(V\): \[ \lambda = \frac{h}{\sqrt{2meV}} \] or approximately \[ \lambda(Å) = \frac{12.27}{\sqrt{V}} \]

Concept:

In Young’s Double Slit Experiment (YDSE), the fringe width \(\beta\) is given by
\[ \beta = \frac{\lambda D}{d} \]

where


\(\lambda\) = wavelength of light
\(D\) = distance between the screen and slits
\(d\) = separation between the slits




Step 1: Effect of medium on wavelength.

When light enters a medium like water with refractive index \(n\), its wavelength changes according to
\[ \lambda' = \frac{\lambda}{n} \]



Step 2: Substitute new wavelength in fringe width formula.
\[ \beta' = \frac{\lambda' D}{d} \]
\[ \beta' = \frac{\lambda D}{nd} \]



Step 3: Compare fringe widths.
\[ \beta' = \frac{\beta}{n} \]

Since \(n > 1\) for water, the fringe width becomes smaller.

Thus, the fringe width decreases. Quick Tip: In YDSE, if the experiment is performed in a medium with refractive index \(n\), the new fringe width becomes \[ \beta' = \frac{\beta}{n} \] Hence fringe width decreases.

Concept:

Maxwell’s equations describe the fundamental laws of electromagnetism. One of these equations states that the divergence of the magnetic field is zero.
\[ \nabla \cdot \vec{B} = 0 \]



Step 1: Interpret the equation.

The divergence of a field measures the net flow of the field out of a point.

If
\[ \nabla \cdot \vec{B} = 0 \]

it means magnetic field lines neither originate nor terminate at any point.



Step 2: Physical implication.

This indicates that magnetic monopoles do not exist. Magnetic field lines always form closed loops.



Step 3: Compare with other Maxwell equations.


Gauss’s law for electricity relates electric field divergence to charge density.
Faraday’s law describes electromagnetic induction.
Ampère–Maxwell law relates magnetic fields to current and changing electric fields.


Thus, the equation representing the absence of magnetic monopoles is
\[ \nabla \cdot \vec{B} = 0 \] Quick Tip: \(\nabla \cdot \vec{B} = 0\) means magnetic field lines form \textbf{closed loops}, implying \textbf{no magnetic monopoles}.

Concept:

An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings.
\[ Q = 0 \]

For an ideal gas undergoing an adiabatic process, the relation between pressure and volume is given by
\[ PV^\gamma = constant \]

where


\(P\) = pressure
\(V\) = volume
\(\gamma = \frac{C_p}{C_v}\) = adiabatic index




Step 1: Use the first law of thermodynamics.
\[ dQ = dU + dW \]

For adiabatic process
\[ dQ = 0 \]
\[ dU = -dW \]



Step 2: Derive the pressure–volume relation.

Using ideal gas relations and thermodynamic identities, the final relation becomes
\[ PV^\gamma = constant \]



Step 3: Interpretation.

During adiabatic expansion, volume increases and pressure decreases more rapidly compared to an isothermal process.



Thus, the correct relation is
\[ PV^\gamma = constant \] Quick Tip: Adiabatic process relations: \[ PV^\gamma = constant, \quad TV^{\gamma-1} = constant \]

Concept:

According to the Bohr model, the electron in a hydrogen atom can occupy only certain discrete energy levels. The energy of the \(n^{th}\) orbit is given by
\[ E_n = -\frac{13.6}{n^2}\ eV \]

where \(n = 1,2,3,\dots\) is the principal quantum number.



Step 1: Identify the ground state.

The ground state corresponds to the lowest possible energy level, which occurs when
\[ n = 1 \]



Step 2: Substitute into the energy formula.
\[ E_1 = -\frac{13.6}{1^2} \]
\[ E_1 = -13.6\ eV \]

Thus, the energy of the ground state of hydrogen is \(-13.6\ eV\). Quick Tip: Energy levels of hydrogen follow \[ E_n = -\frac{13.6}{n^2}\ eV \] The ground state (\(n=1\)) has energy \(-13.6\ eV\).

Concept:

A p-n junction diode contains a depletion region near the junction where mobile charge carriers are absent. This region forms due to recombination of electrons and holes.



Step 1: Effect of reverse bias.

In reverse bias, the positive terminal of the battery is connected to the n-side and the negative terminal to the p-side.

This pulls electrons and holes away from the junction.



Step 2: Resulting change in depletion region.

Since carriers move away from the junction, the depletion region expands and the potential barrier increases.



Step 3: Conclusion.

Therefore, under reverse bias, the width of the depletion layer increases. Quick Tip: Forward bias → depletion region decreases. Reverse bias → depletion region increases.

Concept:

A NAND gate is a digital logic gate that performs the NOT operation on the output of an AND gate.



Step 1: Write the AND gate expression.

For two inputs \(A\) and \(B\),
\[ Y_{AND} = AB \]



Step 2: Apply the NOT operation.

Taking the complement of the AND output,
\[ Y = \overline{AB} \]



Step 3: Interpretation.

The output of a NAND gate becomes \(0\) only when both inputs are \(1\). In all other cases, the output is \(1\).

Thus, the Boolean expression of a two-input NAND gate is
\[ Y = \overline{AB} \] Quick Tip: NAND gate = NOT(AND). Boolean expression: \[ Y = \overline{AB} \] It is also known as a \textbf{universal logic gate}.

Concept:

An inertial frame of reference is a frame in which Newton’s laws of motion hold true. In such frames, objects either remain at rest or move with constant velocity unless acted upon by an external force.



Step 1: Understand the transformation between inertial frames.

If one frame moves with a constant velocity relative to another inertial frame, it also behaves as an inertial frame.



Step 2: Reason for this property.

Because the relative velocity between the frames is constant (no acceleration), Newton’s laws remain valid in both frames.

Thus, a frame moving with constant velocity relative to an inertial frame is also an inertial frame. Quick Tip: All frames moving with \textbf{constant velocity relative to each other} are \textbf{inertial frames}.

Concept:

Polarization is a phenomenon in which the vibrations of a wave are restricted to a particular direction.



Step 1: Understand wave behavior.

Only transverse waves can exhibit polarization because their oscillations occur perpendicular to the direction of propagation.



Step 2: Implication for electromagnetic waves.

When light waves are polarized, it proves that their electric field oscillates in directions perpendicular to the direction of propagation.



Step 3: Conclusion.

Thus, the phenomenon that confirms the transverse nature of electromagnetic waves is polarization. Quick Tip: \textbf{Polarization occurs only in transverse waves}. Its observation proves that light is a \textbf{transverse electromagnetic wave}.

Concept:

Phase velocity \(v_p\) is the velocity with which a single wave phase propagates through space.
\[ v_p = \frac{\omega}{k} \]

where


\(\omega\) = angular frequency
\(k\) = wave number




Step 1: Define group velocity.

Group velocity \(v_g\) is the velocity at which the envelope of a wave packet (or energy) travels.
\[ v_g = \frac{d\omega}{dk} \]



Step 2: Condition for a non-dispersive medium.

In a non-dispersive medium, the wave speed does not depend on frequency.

This means
\[ \omega \propto k \]



Step 3: Resulting relationship.

Under this condition,
\[ v_g = v_p \]

Thus, in a non-dispersive medium, the group velocity equals the phase velocity. Quick Tip: In a \textbf{non-dispersive medium}: \[ v_g = v_p \] In a \textbf{dispersive medium}: \[ v_g \ne v_p \]

Concept:

According to Einstein's Special Theory of Relativity, the length of an object measured in the direction of motion appears shorter to an observer when the object moves at a velocity comparable to the speed of light. This phenomenon is known as length contraction.

The relation for relativistic length contraction is
\[ L = L_0 \sqrt{1-\frac{v^2}{c^2}} \]

where


\(L_0\) = proper length (length in the object's rest frame)
\(L\) = observed length
\(v\) = velocity of the object
\(c\) = speed of light




Step 1: Analyze the velocity dependence.

As \(v\) approaches \(c\),
\[ \frac{v^2}{c^2} \rightarrow 1 \]

Thus,
\[ \sqrt{1-\frac{v^2}{c^2}} \rightarrow 0 \]



Step 2: Conclusion.

Therefore, the observed length \(L\) becomes smaller than the proper length.

Hence, the object appears contracted in the direction of motion. Quick Tip: Relativistic length contraction: \[ L = L_0\sqrt{1-\frac{v^2}{c^2}} \] Objects moving close to the speed of light appear \textbf{shorter along the direction of motion}.

Concept:

Particles with half-integral spin (\(\frac{1}{2}, \frac{3}{2}, \dots\)) are known as fermions. Examples include electrons, protons, and neutrons.

These particles obey the Pauli Exclusion Principle, which states that no two identical fermions can occupy the same quantum state simultaneously.



Step 1: Identify the appropriate statistics.

The statistical distribution that describes the behavior of fermions is the Fermi–Dirac distribution.



Step 2: Write the distribution function.
\[ f(E) = \frac{1}{e^{(E-\mu)/kT} + 1} \]

where


\(E\) = energy of the state
\(\mu\) = chemical potential
\(k\) = Boltzmann constant
\(T\) = absolute temperature




Step 3: Compare with other distributions.


Maxwell–Boltzmann distribution applies to classical particles.
Bose–Einstein distribution describes bosons (integral spin).


Thus, half-integer spin particles follow the Fermi–Dirac distribution. Quick Tip: \textbf{Fermions (half-integer spin)} → Fermi–Dirac statistics \textbf{Bosons (integer spin)} → Bose–Einstein statistics

Concept:

The Quality Factor (Q-factor) of an LCR circuit indicates how sharply the circuit resonates at its resonant frequency.

Mathematically,
\[ Q = \frac{\omega_0 L}{R} \]

or equivalently
\[ Q = \frac{1}{R}\sqrt{\frac{L}{C}} \]

where


\(L\) = inductance
\(C\) = capacitance
\(R\) = resistance
\(\omega_0\) = resonant angular frequency




Step 1: Interpretation of Q-factor.

A high \(Q\)-factor means the circuit has:


Low energy loss
Narrow bandwidth
Sharp resonance peak




Step 2: Bandwidth relation.
\[ Q = \frac{\omega_0}{\Delta \omega} \]

where \(\Delta \omega\) is the bandwidth.



Step 3: Conclusion.

Therefore, the Q-factor describes the sharpness or selectivity of resonance in the circuit. Quick Tip: Higher \(Q\)-factor → sharper resonance and smaller bandwidth. Lower \(Q\)-factor → broader resonance.